* Various "Points"
Battle Arena Points (new), Experience, Gil, Conquest Points, Imperial Sanction Points, Allied Notes, Jettons, Chocobucks, Ballista Points, Walk of Echoes Coins
* Titles
Some people love their titles.
* Achievements
I don't personally play on the Xbox, but I'm sure this is the sort of thing competitive players would want.
* Mog House Furnishings
Trophies, Plaques, Thrones?, Weapon Displays, Armor Displays
* PvP Latent Effect Evoliths and Gear Sets
The appearance of the Aptant gear sets just scream "fantasy team sports" to me. (as well as their customizability)
* Appearance Gear
Gear without combat stats, possibly with fun special features like swimsuits/yukatas/formalwear with teleports/mini-game enhancements. This would include items similar to FanFest Gear - Moogle gear, Chocobo Wand... but different in name/appearance to keep the FanFest gear unique.
* Costume Gear
Chocobo Beret & Goblin Belt types of items (not necessarily those currently available)
~*~ Battle Arena Shop ~*~
All versions of "points", including "Battle Arena Points" can be spent on things in this special shop, so people can pick and chose which games they want to participate in, if any, to buy things from this shop. It will be similar to the shop in FF7's Golden Saucer. Gear in this shop can be used outside of the Arena games. (Conquest Points, Imperial Sanction Points, and other forms of currency can also be used in place of, or perhaps be exchanged for, Battle Arena Points.)
Conqueror Gear - New Ballista/Brenner Aptant sets (Some sets will have choosable "Augments", like the 3 add-on rewards with stats only during PvP games, some will have many Neo-evolith slots and some sets will just be for looks and have no evolith slots, to be used in creating unique "Uniforms".)
Neo-evoliths - Evoliths without the massive restrictions of "tiny bonus only against monster family X" or "During a full eclipse on Firesday, while under the effect of terror." There are other threads (I may add links) on how to make Evoliths useful, so I won't go into that here. Just don't think I mean the current evolith system as it stands, because that's just crazy. (Some Evoliths may have the restriction of "Must be participating in a PvP contest" as mentioned above)
Chocobo Knight Gear - Stats aimed at modifying chocobos in Chocobo Derby games - and maybe remove (not just extend) the chocobo ride time duration limits outside the games.
Wild Rider Gear - Stats aimed at enhancing non-Chocobo mounts in Chocobo Derby games - and maybe call new mounts outside of Chocobo Derby.
Teleportation items - Nation Stables Collar/Medal/Scarf, Crag Rings, Teleport Earrings, Gate Glyphs
"New Merits" - This could be one method to obtain new weaponskills (possibly gain a special currency by fighting the Crystal War generals who use special weaponskills to buy only the weaponskill they use), new job traits and new job abilities. Not only gear can be sold in the shop... this would be associated with the new merit system to break through the level 99 limits, since the Simulacrum is in a very similar situation to that which players who have reached level 99 in multiple jobs are in, having mastered all there is to master. The hatred some people have toward mini-games, and the fact that not everyone will own this expansion, would require this to simply be an alternate way to accomplish the same goals as those without the expansion or who don't want to participate in mini-games. (for instance, I'd never participate in Ballista/Brenner/Diorama, even if it produced the ultimate weapons in the game, but I'd participate in other parts of the arena every single day, all day, if given the chance.)